Subgroups — #13127736

Order 1664.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719244	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#442	C₂	1	1	yes	maximal, characteristic
4	#11718813	C₂	1	1	yes	maximal, characteristic
5	D₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	#11718670	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718669	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718517	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718516	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718427	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718422	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718422	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718420	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718363	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₂₆	—	2	1	no	—
17	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	#88178	—	2	1	no	—
25	#88178	—	2	1	no	—
26	C₂×#6101	—	2	1	no	—
27	#88144	C₂²	1	1	yes	characteristic
28	#88142	C₂²	1	1	yes	characteristic
29	#88142	C₂²	1	1	yes	characteristic
30	#88142	C₂²	1	1	yes	characteristic
31	#88132	—	2	1	no	—
32	#88132	—	2	1	no	—
33	#88132	—	2	1	no	—
34	#88132	—	2	1	no	—
35	#88124	C₂²	1	1	yes	characteristic
36	#88123	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
38	C₁₃×#106	C₂²	1	1	yes	characteristic
39	C₁₃×#102	C₂²	1	1	yes	characteristic
40	C₁₃×#102	C₂²	1	1	yes	characteristic
41	C₁₃×#97	C₂²	1	1	yes	characteristic
42	#88072	C₂²	1	1	yes	characteristic
43	#88072	C₂²	1	1	yes	characteristic
44	#88072	C₂²	1	1	yes	characteristic
45	#88072	C₂²	1	1	yes	characteristic
46	#88058	C₂²	1	1	yes	characteristic
47	#88058	C₂²	1	1	yes	characteristic
48	#88057	C₂²	1	1	yes	characteristic
49	#88057	C₂²	1	1	yes	characteristic
50	#88055	C₂²	1	1	yes	characteristic
51	#88054	C₂²	1	1	yes	characteristic
52	#88054	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
54	#88048	C₂²	1	1	yes	characteristic
55	#88048	C₂²	1	1	yes	characteristic
56	#88048	C₂²	1	1	yes	characteristic
57	#88047	C₂²	1	1	yes	characteristic
58	C₄⋊#1055	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	—	2	1	no	—
65	C₂²×Dic₁₃	—	2	1	no	—
66	C₂²×Dic₁₃	—	2	1	no	—
67	C₂×Dic₂₆	—	2	1	no	—
68	C₂×Dic₂₆	—	2	1	no	—
69	C₂×Dic₂₆	—	2	1	no	—
70	C₂×Dic₂₆	—	2	1	no	—
71	C₂×Dic₂₆	—	4	4	no	—
72	C₂×Dic₂₆	D₄	1	1	yes	characteristic
73	C₁₃×D₈	Q₈	1	4	yes	complemented
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₁₃×C₂²⋊C₄	—	2	1	no	—
79	C₁₃×C₂²⋊C₄	—	2	1	no	—
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	C₂²⋊Dic₁₃	—	2	1	no	—
83	C₂²⋊Dic₁₃	—	2	1	no	—
84	C₂²⋊Dic₁₃	—	2	1	no	—
85	C₄⋊Dic₁₃	—	2	1	no	—
86	C₄⋊Dic₁₃	—	2	1	no	—
87	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
88	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
92	#6101	—	2	1	no	—
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	#6101	—	2	1	no	—
98	#6101	D₄	1	1	yes	characteristic
99	#6101	D₄	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
102	C₂×#1055	C₂³	1	1	yes	characteristic
103	C₂×#1055	C₂³	1	1	yes	characteristic
104	#3384	—	13	1	no	maximal, Sylow
105	C₂³×C₁₃	—	2	1	no	—
106	C₂³×C₁₃	—	2	1	no	—
107	D₄×C₁₃	—	2	1	no	—
108	D₄×C₁₃	—	2	1	no	—
109	D₄×C₁₃	C₂×Q₈	1	2	yes	—
110	D₄×C₁₃	C₂×Q₈	1	2	yes	complemented
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	8	1	no	—
115	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	1	no	—
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	8	1	no	—
126	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
130	Dic₂₆	—	2	2	no	—
131	Dic₂₆	—	4	2	no	—
132	Dic₂₆	—	4	2	no	—
133	Dic₂₆	—	4	2	no	—
134	Dic₂₆	—	8	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	C₂×Q₈	1	2	yes	complemented
137	#1055	—	2	1	no	—
138	#1055	—	2	1	no	—
139	#554	—	13	1	no	—
140	D₄×Q₈	—	13	1	no	—
141	#501	—	13	1	no	—
142	#501	—	13	1	no	—
143	#498	—	13	1	no	—
144	#478	—	13	1	no	—
145	#478	—	13	1	no	—
146	#476	—	13	1	no	—
147	#476	—	13	1	no	—
148	#476	—	13	1	no	—
149	#474	—	13	1	no	—
150	#446	—	13	1	no	—
151	#442	—	13	1	no	—
152	#442	D₁₃	1	1	yes	characteristic
153	C₄×D₈	—	13	1	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	4	1	no	—
164	C₄×C₁₃	—	4	1	no	—
165	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
166	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
167	Dic₁₃	—	2	1	no	—
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	—	4	1	no	—
171	Dic₁₃	—	4	1	no	—
172	Dic₁₃	—	4	1	no	—
173	Dic₁₃	—	4	1	no	—
174	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
175	C₂²×Q₈	—	26	1	no	—
176	C₂×D₈	C₂×D₁₃	1	1	yes	characteristic
177	C₄⋊Q₈	—	13	1	no	—
178	C₄⋊Q₈	—	13	1	no	—
179	C₄⋊Q₈	—	13	1	no	—
180	C₄⋊Q₈	—	13	1	no	—
181	#125	—	13	1	no	—
182	#125	—	13	1	no	—
183	C₂²⋊Q₈	—	26	1	no	—
184	C₂²⋊Q₈	—	26	1	no	—
185	C₂²⋊Q₈	—	26	1	no	—
186	C₂²⋊Q₈	—	26	1	no	—
187	C₂²⋊Q₈	—	26	1	no	—
188	C₂²⋊Q₈	—	26	1	no	—
189	C₄×Q₈	—	13	1	no	—
190	C₄×D₄	—	13	1	no	—
191	C₄×D₄	—	13	1	no	—
192	C₄×D₄	—	13	1	no	—
193	C₄×D₄	—	13	1	no	—
194	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
195	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	26	1	no	—
197	#107	—	13	1	no	—
198	#107	—	13	1	no	—
199	#107	—	13	1	no	—
200	#106	—	13	1	no	—
201	#106	—	13	1	no	—
202	#106	—	13	1	no	—
203	#106	—	13	1	no	—
204	#106	—	13	1	no	—
205	#106	C₂×D₁₃	1	1	yes	characteristic
206	C₄⋊C₈	—	13	1	no	—
207	C₄⋊C₈	—	13	1	no	—
208	C₄⋊C₈	—	13	1	no	—
209	#102	—	13	1	no	—
210	#102	—	13	1	no	—
211	#102	—	13	1	no	—
212	#102	—	13	1	no	—
213	#102	C₂×D₁₃	1	1	yes	characteristic
214	#102	C₂×D₁₃	1	1	yes	characteristic
215	#97	—	13	1	no	—
216	#97	C₂×D₁₃	1	1	yes	characteristic
217	C₄×C₈	—	13	1	no	—
218	C₂×C₁₃	—	4	2	no	—
219	C₂×C₁₃	—	4	2	no	—
220	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
221	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×C₁₃	#577	1	1	yes	characteristic
223	C₂×Q₈	—	13	1	no	—
224	C₂×Q₈	—	26	1	no	—
225	C₂×Q₈	—	26	1	no	—
226	C₂×Q₈	—	26	1	no	—
227	C₂×Q₈	—	26	1	no	—
228	C₂×Q₈	—	52	4	no	—
229	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	—	26	1	no	—
235	C₂²×C₄	—	26	1	no	—
236	C₂²×C₄	—	26	1	no	—
237	D₈	Dic₂₆	1	4	yes	complemented
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	13	1	no	—
243	C₄⋊C₄	—	13	1	no	—
244	C₄⋊C₄	—	13	1	no	—
245	C₄⋊C₄	—	13	1	no	—
246	C₄⋊C₄	—	13	1	no	—
247	C₄⋊C₄	—	13	1	no	—
248	C₄⋊C₄	—	13	1	no	—
249	C₄⋊C₄	—	26	1	no	—
250	C₄⋊C₄	—	26	1	no	—
251	C₄⋊C₄	—	26	1	no	—
252	C₄⋊C₄	—	26	1	no	—
253	C₄⋊C₄	—	26	1	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	—	26	1	no	—
257	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	26	1	no	—
262	C₂²⋊C₄	—	26	1	no	—
263	C₂²⋊C₄	—	26	1	no	—
264	C₂²⋊C₄	—	26	1	no	—
265	C₄²	—	13	1	no	—
266	C₄²	—	13	1	no	—
267	C₄²	C₂²×D₁₃	1	1	yes	characteristic
268	C₁₃	#3384	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	Q₈	—	26	2	no	—
272	Q₈	—	52	2	no	—
273	Q₈	—	52	2	no	—
274	Q₈	—	52	2	no	—
275	Q₈	—	104	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	C₂×Dic₂₆	1	2	yes	—
279	D₄	C₂×Dic₂₆	1	2	yes	complemented
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	26	1	no	—
289	C₂×C₄	—	26	1	no	—
290	C₂×C₄	—	26	1	no	—
291	C₂×C₄	—	26	1	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	—	104	1	no	—
296	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	26	1	no	—
301	C₈	—	26	1	no	—
302	C₈	C₂×Dic₂₆	1	2	yes	complemented
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	13	2	no	—
315	C₄	—	26	1	no	—
316	C₄	—	26	1	no	—
317	C₄	—	26	1	no	—
318	C₄	—	52	1	no	—
319	C₄	—	52	1	no	—
320	C₄	—	52	1	no	—
321	C₄	—	52	1	no	—
322	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
323	C₄	#88257	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	4	2	no	—
326	C₂	D₄×Dic₂₆	1	1	yes	characteristic
327	C₂	D₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂	#11719480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#442	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×D₈	C₂²
hypercenter	#442	D₁₃
hypocenter	C₁₃	#3384
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #11719244	C₂	semidirect
2. D₄×Dic₂₆	C₂	semidirect
6. #11718670	C₂	semidirect
7. #11718669	C₂	semidirect
8. #11718517	C₂	semidirect
9. #11718516	C₂	semidirect
10. #11718427	C₂	semidirect
11. #11718422	C₂	semidirect
12. #11718422	C₂	semidirect
13. #11718420	C₂	semidirect
14. #11718363	C₂	semidirect
15. #11718362	C₂	semidirect
73. C₁₃×D₈	Q₈	semidirect
110. D₄×C₁₃	C₂×Q₈	semidirect
136. C₈×C₁₃	C₂×Q₈	semidirect
237. D₈	Dic₂₆	semidirect
268. C₁₃	#3384	semidirect
279. D₄	C₂×Dic₂₆	semidirect
302. C₈	C₂×Dic₂₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11719244	2
D₄×Dic₂₆	2
C₁₃×#442	2
#11718813	2
D₈×Dic₁₃	2
#11718670	2
#11718669	2
#11718517	2
#11718516	2
#11718427	2
#11718422	2
#11718420	2
#11718363	2
#11718362	2
#3384	13